Introduction to Probability and Statistics for Engineers and Scientists

358 Chapter 9: Regression

Using Equation 9.3.1 along with the relationship

A=

∑n

i= 1

Yi

n

−Bx

shows thatAcan also be expressed as a linear combination of the independent normal
random variablesYi,i=1,...,n and is thus also normally distributed. Its mean is
obtained from

E[A]=

∑n

i= 1

E[Yi]

n

−xE[B]

=

∑n

i= 1

(α+βxi)

n

−xβ

=α+βx−xβ

=α

ThusAis also an unbiased estimator. The variance ofAis computed by first expressing
Aas a linear combination of theYi. The result (whose details are left as an exercise) is that

Var(A)=

σ^2

∑n

i= 1

xi^2

n

(n

∑

i= 1

xi^2 −nx^2

) (9.3.3)

The quantitiesYi−A−Bxi,i=1,...,n, which represent the differences between the
actual responses (that is, theYi) and their least squares estimators (that is,A+Bxi) are
called theresiduals. The sum of squares of the residuals

SSR=

∑n

i= 1

(Yi−A−Bxi)^2

can be utilized to estimate the unknown error varianceσ^2. Indeed, it can be shown that

SSR

σ^2

∼χn^2 − 2

That is,SSR/σ^2 has a chi-square distribution withn−2 degrees of freedom, which implies
that

E

[

SSR

σ^2

]

=n− 2